ON THE PERFORMANCE OF HOPFIELD NETWORK FOR GRAPH SEARCH PROBLEM

Transcription

1 ON THE PERFORMANCE OF HOPFIELD NETWORK FOR GRAPH SEARCH PROBLEM Gursel Serpen and Azadeh Parvin phone: (49) fax: (49) The University of Toledo Electrical Engineering & Computer Science Department Toledo, OH USA

2 ABSTRACT This paper presents a study on the performance of the Hopfield neural network algorithm for the graph path search problem. Specifically, performance of the Hopfield network is studied from the dynamic systems stability perspective. Simulations of the time behavior of the neural network is augmented with exhaustive stability analysis of the equilibrium points of the network dynamics. The goal is to understand the reasons for the well-known deficiency of the Hopfield network algorithm: the inability to scale up with the problem size. A recent procedure, which establishes solutions as stable equilibrium points in the state space of the network dynamics, is employed to define the constraint weight parameters of the Hopfield neural network. Simulation study of the network and stability analysis of equilibrium points indicate that a large set of nonsolution equilibrium points also become stable whenever constraint weight parameters are set to make the solution equilibrium points stable. As the problem size grows, the number of stable non-solution equilibrium points increases at a much faster rate than the number of stable solution equilibrium points: the network becomes more likely to converge to a non-solution equilibrium point. Keywords - neural networks, Hopfield networks, weight parameters, stability analysis, dynamic system, optimization, constraint satisfaction

3 3. Introduction Hopfield networks have been employed as fixed-point attractors to solve a large set of constraint satisfaction and optimization problems [3, 4, 5, 6, 7]. Time-behavior of the network can be succinctly expressed as performing a gradient descent search in the Lyapunov function space, providing a measure for the degree of satisfying the set of constraints associated with a given problem [6]. Main promise of the Hopfield network is to converge to the stable fixed-point located within the basin of attraction implied by the initial conditions of the network dynamics. The Hopfield network can successfully address certain constraint satisfaction or optimization problems for which a local optimum solution is acceptable [,, 5]. A Boltzmann machine, a stochastic version of the Hopfield neural network algorithm, is employed if the global optimum solution is desired []. When the Hopfield network is employed in a fixed-point attractor mode, the neural network design task incorporates establishing the solutions of a given optimization problem as local minimum points or equivalently stable points in the Lyapunov space or state space, respectively. This task has been performed in an ad hoc manner until recently since the original Hopfield algorithm did not propose a way to define the constraint weight parameters [,, 3, 4, 7, 8, 9,, 3, 4]. In most cases, constraint weight parameters were set using empirical guidelines. As a result, stability properties of solution equilibrium points could not be determined. Recently, Serpen [9] and Abe [] have devised procedures, employing two different approaches, to set the constraint weight parameters to establish solutions as stable equilibrium points in the state space of the problem dynamics. Most studies in the literature consider the Traveling Salesman Problem (TSP), which is NPcomplete, as the benchmark problem for performance analysis of the Hopfield network algorithm. In an earlier work [8], it was shown that the Hopfield network converged to a solution of the TSP after each relaxation when the constraint weight parameters were set in accordance with the procedure specified as in Serpen [9]. However, the solution quality was, at best, locally optimum. A state space analysis indicated that all solutions were stable and no other equilibrium point was stable. In summary, the Hopfield network always located a solution for the TSP but the quality of solutions, in terms of the overall travel distance, was average. On the other hand, initial work with the path search problem in graphs indicated that the

4 4 Hopfield network with the constraint weight parameters defined as in Serpen [9] often failed to converge to a solution. The graph path search problem (GPSP) proved to be "hard" for the Hopfield network paradigm. The goal of this paper is to investigate the reasons for the poor performance of the Hopfield network on the GPSP. Towards that goal, an extensive set of simulation and mathematical stability analysis will be performed. Specifically, stability of solution points, the set of stable equilibrium points, and convergence properties of the Hopfield network algorithm will be studied. The structure of the paper is as follows. The discrete Hopfield network will be presented in Section. Definitions of GPSP, network topology, and energy function and bounds on constraint weight parameters are given in Section. Simulation study and mathematical stability analysis are presented in Section 3. Conclusions will be presented in Section 4.. List of Symbols Presented below is a list of symbols employed in the remainder of the paper. s i w ij b i u i θ i E net i C ϕ output of network node i weight between nodes i and j external bias term for node i state of node i threshold of node i energy or Liapunov function network input for node i constraint ϕ g ϕ weight parameter for C ϕ δ ϕ ij d ij ϕ function representing interaction between nodes i and j under C ϕ function representing the cost of interaction between nodes i and j under C ϕ

5 5. Definitions A list of definitions which will be employed in the remainder of the paper is presented below. Definition. The state space set contains all N N-bit binary vectors for an N-node network. Definition. The stable point set includes the binary vectors which are stable points of the Hopfield network dynamics for a given problem. Definition 3. The solution set contains those N-bit binary vectors for an N-node network which are solutions of a problem. Definition 4. The stable solution set consists of those solutions of a given problem which are stable points of the network dynamics. Definition 5. A relaxation (iteration) of the Hopfield network is the total computation effort for the network to start from an initial state and to converge to a final state. Definition 6. The convergence rate is computed by the number of convergences to solutions divided by the number of total relaxations attempted. Definition 7. The convergence ratio is calculated by dividing the number of elements in the stable solution set to the number of elements in the stable point set. Definition 8. An operating point is an instance of the values for the set of constraint weight parameters. Definition 9. For an N N array network topology, row(i) and col(i) are equal to the row and column indices of the node si with i =,,..., N, respectively. Definition 0. A constraint is called hard if violating it necessarily prevents the network from finding a solution. Definition. A soft constraint is employed to map a cost measure associated with the quality of a solution as typically found in optimization problems..3 The Discrete Hopfield Network The discrete Hopfield network [8, 9, 0, ] is a nonlinear dynamic system with the following formal definition. Let s i represent a node output where s i = 0, for i =,,, N and N is the number of network nodes. Then, the equation given by

6 6 N N E = wijsis j bisi + Θ isi, i j () i= j= i= i= is the Liapunov function whose local minima are the final states of the network with node dynamics defined by s N i k i k i s N + = 0 if net < Θ, i k i k i + = if net > Θ, and i k i k i k i s + = s if net = Θ, i =,,, N, () where k is a discrete time index, N neti k = wijs k j + bi (3) j= with i j and Θ i is the threshold of node s i. The weight term is defined by Z ij = ϕ ϕ ϕ ij ij ϕ= w g δ d ϕ, g R + if the where Z is the number of constraints. Given the set of constraints C { C, C,, } hypotheses nodes s i C Z and s represent for C ϕ are mutually supporting and g R if the same hypotheses j ϕ ϕ are mutually conflicting. The term δ ϕ ij is equal to if the two hypotheses represented by nodes s s i and j are related under C ϕ and is equal to 0 otherwise. The d ij ϕ term is equal to for all i and j under a hard constraint and is a predefined cost for a soft constraint, which is typically associated with a cost term in optimization problems.

7 7. Graph Path Search Problem A graph with directed edges is used as a mathematical model for a number of real-life problems [5]; example problems include the path planning in a task state space, the nonlinear multi-commodity flow problem and the routing problem in computer networks. Determining a solution to those problems often requires computation of the shortest path between two vertices of a graph: the source vertex and the target vertex. A directed graph is a set of m vertices, Vi, i =,,, m, and k ( = m ) directed edges, eij, i, j =,,..., m, where some of the edges may not exist. Each directed edge in the graph has an associated length or cost. The length of an edge is represented with a real number. A path between a source vertex, V s, and a target vertex, V t, is an ordered sequence of non-zero length edges. The path length is given by the algebraic sum of lengths of all the edges in that path. A solution for the GPSP is then to identify the path which has the minimum length. For the case where all edges have the same length, the shortest path is equivalent to the minimum number of edges which connect the source vertex to the target vertex. The scope of this paper is limited to directed graphs with non-weighted edges, which will be adequate for the purposes of this study. From the graph theoretic viewpoint, the shortest path between two vertices of a non-weighted directed graph is defined as a sub-graph which meets all of the following criteria: The sub-graph representing a path is both asymmetric and irreflexive. Each vertex, except the source and target vertices, must have in-degree of one and out-degree of one. The source vertex has in-degree of zero and out-degree of one. The target vertex has in-degree of one and out-degree of zero. The length of the shortest path is equal to the power of the adjacency matrix which has the first nonzero entry in the row and column locations as defined by the source and target vertices, respectively. Given the graph-theoretic constraints a path specification has to satisfy, the next step is to define the corresponding topological constraints of the GPSP. The topology of the network is modeled after the

8 8 adjacency matrix of a given graph, where each network node represents a graph edge. An active node at row r and column c indicates that the edge from vertex V r to vertex V c is included in the path specification. Since the shortest path is irreflexive, the nodes located along the main diagonal are clamped to zero as these nodes represent the hypothesis that the shortest path has a self-loop for vertex V i. In order to map the in-degree of zero constraint for the source vertex, the nodes in the column labeled by the source vertex are clamped to zero. Similarly, the nodes in the row labeled by the target vertex are clamped to zero to enforce the out-degree of zero constraint for the target vertex. The nodes in the locations where there is a zero entry in the associated adjacency matrix are also clamped to zero, since a zero entry in the adjacency matrix implies that the graph does not have an edge between the related vertices. An adjacency matrix has N entries for an N-vertex digraph. Given that certain nodes are clamped to zero, the network will have K, K < N, unclamped nodes. Note that the clamped nodes need not be included in the computations associated with the network simulations.. Definition of the Energy Function The asymmetry of a graph which represents the shortest path requires that one of the two entries, at most, located at symmetric positions with respect to the main diagonal of the adjacency matrix, be equal to one. Any two nodes in the network interact if the row index of node s i equals the column index of node s j and the column index of node s i equals the row index of node s j. Since only one of two interacting nodes can be equal to, the type of interaction between the nodes is inhibitory, g a R. The energy term for this inhibitory constraint is of the form: K Ea = gaδ ij a sis j, i= K j= where δ a ij = if row ( i) = col( j) and col( i) = row( j) ; otherwise δ ij a = 0 for i, j =,,..., K and i j. The subscript/superscript a indicates the asymmetry inhibition. Consider the network nodes in rows and columns not associated with the source or target vertices. If a vertex is included in the shortest path specification, then there is exactly one active node in both the row

9 9 and the column labeled by this vertex. On the other hand, there are no active nodes within the rows and columns labeled by vertices which do not belong to the path specification. Thus, when the network converges to a solution, there can be at most one node active per row or column. A digraph vertex, belonging to the path specification and having in-degree of one, implies the existence of a single in the associated column of the adjacency matrix. The condition for two nodes to interact under the column constraint is that the column index of node s i equals the column index of node s j. Similarly, a digraph vertex with an out-degree of one requires exactly a single to exist in the associated row of the adjacency matrix. Two nodes interact under the row constraint if the row index of node s i equals the row index of node s j. These constraints permit at most one of the nodes in a column or row to be active at any time; thus the type of interaction is inhibitory with g, g R. The energy term for the column inhibition constraint is defined by c r K Ec = gc δ ij c sis j, i= K j= c where = col( i) = col( j) δ ij if ; otherwise δ c ij = 0 for i, j =,,..., K and i j given that both col( i) and col( j) do not correspond to the target vertex column. Similarly, the energy term for row inhibition constraint is defined by Er = gr δ ijs s K i= K j= r i j r where = row( i) = row( j) δ ij if ; otherwise δ r ij = 0 for i, j =,,..., K and i j given that both row( i) and row( j) are not the source vertex row. Subscripts/superscripts r and c indicate the row and the column inhibition constraints, respectively. Since the source vertex must have out-degree of one, there is exactly one node active within the source vertex row for a solution array. Given the target vertex has in-degree of one, exactly one node is

10 0 active within the target vertex column for a solution array. These two constraints can be mapped employing the energy terms defined by Est = - g r - skl - gc - smn, l m where k represents the source vertex row and l is the index for the columns of the network in the first energy term, n represents the target vertex column and m is the index for the rows of the network in the second energy term, and gc R and gr R are the constraint weight parameters associated with the column and the row inhibition constraints, respectively. The in-degree and out-degree values for vertices of a solution graph depend on if a particular vertex belongs to the shortest path specification. Any vertex other than the source and the target vertices has both the in-degree and the out-degree equal to one if that vertex belongs to the path specification. The source vertex has in-degree of zero and out-degree of one. The target vertex has in-degree of one and outdegree of zero. A vertex not in the path specification has both the in-degree and the out-degree equal to zero. In terms of the adjacency matrix, a vertex with both the in-degree and the out-degree equal to one implies a in both the row and the column labeled by this vertex. In general, if there exists a in a particular row r and column c, then there must exist a in the corresponding column r and row c of the solution array. This constraint which enforces both the in-degree and the out-degree of a vertex within the path specification to be equal to will be decomposed into two sub-constraints, which are called the subconstraint column-to-row excitation and the sub-constraint row-to-column excitation, for ease of mapping to the network topology. The sub-constraint column-to-row excitation states that if the in-degree of vertex V i is one, then the out-degree of the same vertex must also be one. For the sub-constraint row-to-column excitation, if the outdegree of vertex V i is one, then the in-degree of the same vertex must also be one. In terms of the network topology, if the column sum associated with the vertex V i is equal to one, then the row sum associated with the same vertex must also be one for the sub-constraint column-to-row excitation. The nodes in the target vertex column do not belong to the interaction topology of this sub-constraint since the out-degree of the

11 target vertex is set to zero. Similarly, if the row sum associated with the vertex V i is one, then the column sum associated with the same vertex must be one for the sub-constraint row-to-column excitation. The nodes in the source vertex row do not interact under the sub-constraint row-to-column excitation since the nodes in the source vertex column are clamped to zero to set the in-degree of the source vertex to zero. Thus, two nodes, s i and s j, interact under sub-constraint column-to-row excitation if the column index of s i is equal to the row index of s j, and s i is not located within the target vertex column because the nodes in the target vertex row are clamped to zero. Similarly, two nodes, s i and s j, interact under subconstraint row-to-column excitation if the row index of s i is equal to the column index of s j and s i is not located within the source vertex row, since the nodes within the source vertex column are clamped to zero. The nodes of a given a row and column pair interact under the sub-constraint column-to-row excitation or the sub-constraint row-to-column excitation. Sub-constraint column-to-row excitation can be mapped by E cr = gd sr c - s cc + - s r c s cc, 4 c r c r c where r is the index for the rows and c, c are the indices for the columns of the network topology with the property that both c and c are not the target vertex column. Similarly, sub-constraint row-to-column excitation can be mapped by E rc = g d sr c - s r r + - s r c s rr, 4 r c r c r where c is the index for the columns and r, r are the indices for the rows of the network topology with the property that both r and r are not the source vertex row. These energy terms have a value of zero if the corresponding column/row sums are both equal to zero or both equal to one. The values of the energy terms are greater than zero if a column/row sum is zero

12 and the corresponding row/column sum is greater than zero. On the other hand, the energy term values become less than zero if a column/row sum is equal to one and the corresponding row/column sum is greater than one. Although the energy terms favor each of the interacting columns and rows having more than one node active, other constraints force the network dynamics to favor row and column sums with a value of one. A single constraint weight parameter for both sub-constraints, g d R +, will be employed given that two sub-constraints are obtained by decomposing the original constraint. Any solution path must be of minimum length. The length of the shortest path can be computed without knowing the actual path itself. The global inhibition constraint can be used to force the network to have M-out-of-K nodes active. The value of M is equal to the length of the shortest path less, since the nodes within the source vertex row and the target vertex column do not interact under this constraint and a solution array has one active node within both the source vertex row and the target vertex column. The following energy term is minimum when exactly M nodes are active, E = gγ si - M i γ, i =,,..., K, where row( i) and col( i) are not the source vertex row and the target vertex column, respectively. The energy function for the network is the algebraic sum of all individual energy terms and is given by E = Ea + Er + Ec + Est + Erc + Ecr + E γ. In accordance with the work by Serpen [9], the applicable bound on constraint weight parameters for solutions of the GPSP to be stable is given by g + g gγ g. (4) r c d Note that this inequality establishes a relationship between only the magnitudes of the constraint weight parameters.

13 3 3. Simulation Analysis The identification of set ordering relationships between two sets of interest, the solution set and the stable point set, is necessary to understand the convergence characteristics of the Hopfield network. The solution set represents all those output vectors that satisfy the constraints of a given problem and the stable point set consists of all those output vectors that are stable equilibrium points in the state space of the Hopfield network dynamics. It is important to observe that if the solution set is equal to the stable point set, then the Hopfield network will always converge to a solution point after each relaxation. Any other relationship between these two sets will either cause some solution points to be unstable (in the case where the stable point set is a proper subset of the solution point set) or some non-solution points to be stable (in the case where the solution point set is a proper subset of the stable point set). The latter is the only feasible case for the bounds given by Equation 4 since, by definition, bounds on the constraint weight parameters, as suggested by Abe [] and Serpen [9], establish stability of all solution points. 3. Description of the Simulation Study and Testing Methodology Evaluation of the network performance is realized by employing two techniques: simulation based relaxation study and mathematical stability analysis. The simulation based relaxation study is used to observe the time behavior of the network dynamics. Specifically, the rate at which the network converges to fixed points is used as the measure for the network performance. Simulation based relaxation study provides a statistical estimate of the convergence rate. Stability analysis of equilibrium points in the problem state space is used to identify the set of stable points. A second measure of network performance, the convergence ratio, is computed by algebraically dividing the number of stable solution points by the number of stable equilibrium points, using the findings of the mathematical stability analysis. Two network performance measures, the convergence rate and the convergence ratio, are expected to correlate to a very large degree by definition. Stability analysis of all equilibrium points in the problem state space is not computationally feasible for large problem sizes. An K-node network with discrete node dynamics will have K states in the K dimensional state space. This large number indicates that the upper limit for the problem size is the 5 5

14 4 node network for problems which require two dimensional arrays as network topologies, where no nodes are clamped to either zero or one. The parameter set employed in the study included the graph size, the connectivity of graph and the path length. Instances of the directed graphs which were employed in the simulation study and in the stability analysis were created by modifying: ) the number of vertices in the directed graph, ) the connectivity of the digraph, which is the ratio of the number of edges in the graph to the number of edges in the same graph when it is fully connected, and 3) the path length between the source and the target vertices of the digraph. Graph edges were randomly defined in order to establish the generality of the simulation results. A random variable uniform in the interval [0,], ρ, was used. An edge e i was included in the graph specification if the following inequality held: ρ i > Connectivity Level. To test the performance of the network, the number of vertices, the connectivity level and the desired path length were provided to the algorithm. The length of a path between any two pairs of vertices of the digraph was computed using the adjacency matrix, although the path itself was not known. A path length of less than three was not used in any of the test cases. The operating points for the network were generated using Equation 4. Relative magnitudes of constraint weight parameters rather than their absolute values are manipulated to generate a complete set of operating points which are presented in Table. Constraint Weight Operating Point Parameters gr = gc = ga g d g γ Table. Operating point definitions for the GPSP.

15 5 3. Simulation Results An initial evaluation of the network performance indicated that the convergence rate was less than 00% and varied significantly as the graph size, the operating point, the path length, the connectivity level, and the graph instances differed. Therefore, structured tests were run to better understand the relationship between the convergence rate and the variables in the parameter set. Additionally, the dependencies between the variables in the parameter set was also taken into consideration and the simulation study was modified accordingly. An example of this is the dependence of the path length on the graph size, the connectivity level, and the graph instance, where the path length decreases as the connectivity level increases. In the tests that follow, three out of four variables (the graph size, the connectivity level, the path length, and the operating point) are fixed and the convergence rate is observed as the fourth one varies. The fifth variable, the graph instance, can not be controlled since the graph edges are randomly created. The first test is run for the case where the operating point, the graph size, and the path length are fixed and the connectivity level is varied. The network performance strongly depends on the connectivity level as given in Figure. Specifically, the convergence rate and the convergence ratio decrease as the connectivity increases: the convergence rate and ratio drops from 50-60% to about 5% as the connectivity increases from 0. to 0.8. There is a high level of correlation between the convergence ratio and the convergence rate as the connectivity varies (the correlation coefficient is ). The decrease in the convergence ratio indicates that the cardinality of the stable solution set becomes much smaller than that of the stable solution set as the connectivity increases. As a result, the network does not always converge to a solution.

16 6 Performance vs. Connectivity (6-vertex, path length of 3, operating point 7) Convergence Rate Ratio Connectivity Figure. Performance analysis at various connectivity levels for the GPSP. The second test case involved the evaluation of the network performance as the operating point varied. This test was performed with the following parameter values: a digraph size of six vertices, a path length of 3, and a connectivity of 0.6. The same graph instance was used for all experiments. The size of the state space set and the number of solutions for this graph instance were 4096 and 3, respectively. Results presented in Figure indicate that the network performance significantly varied as the operating point changed. Operating point guided the network towards an almost 70% convergence rate while operating point 5 caused the network performance to drop down to approximately 0% convergence rate. Additionally, the convergence ratio is not highly correlated with the convergence rate. It is tempting to infer that initialization of the network and the shape and size of the basins of attraction associated with the stable equilibrium points played a significant role in the lack of correlation. Exhaustive stability evaluation of equilibrium points demonstrated that all solutions are stable.

17 7 Performance vs. Operating Point (6-vertex, path length of 3, connectivity 0.6) Convergence Operating Point Rate Ratio Figure. Performance analysis at various operating points for the GPSP. The third test case involved studying the variation of the convergence rate and the convergence ratio with respect to the variation in the graph size. In simulation studies, a graph connectivity level of 0.3 and a path length of half the number of vertices were used. For the graphs with an odd number of vertices, the path length was determined by rounding the computed value up to the next larger integer. The operating point employed in the experiments was number 8 in Table. Simulation results, presented in Figure 3, indicate that the network performance degraded drastically (convergence rate drops from 70% to about 0% for a minimal increase in the vertex count from 6 to 0) as the graph size increased. Similarly, the convergence ratio dropped from 30% to % for the same increase in the graph size, which is highly correlated with the variations in the convergence rate. Performance vs. Problem Size (operating point 8, connectivity 0.3) Convergence Rate Ratio Vertex Count Figure 3. Performance analysis for various GPSP sizes.

18 8 The next test case was set up to observe the change in the convergence rate with respect to the changes in the path length and the connectivity level for a 0-vertex graph. The connectivity level and the path length varied in the ranges [0., 0.8] and [4, 8], respectively. Operating point number 8 in Table was employed for this test case. The results are summarized in Table. The symbol " " in a box located at row r and column c means a 0-vertex graph with a path length of c and a connectivity of r could not be generated in 00 attempts. Connectivity Path Length Level Table. Convergence rate vs. path length connectivity level for the GPSP. The data in Table demonstrates that the convergence rate is significantly higher for low values of the graph connectivity or the path length. As an example, the convergence rate is 56% for a connectivity level of 0.3 and a path length of 4. The convergence rate drops to % for a path length of 8 and the same connectivity level. Simulation results presented in Table correlate with findings in Figures and 3 to a large degree. Exhaustive stability analysis was not conducted for this test since the 0-vertex problem with 0.8 connectivity level generates approximately bit binary vectors to evaluate. Another study analyzed the performance of the network with respect to variations in the graph instances. A graph with 0 vertices, 0.3 connectivity, and a path length of 5 was employed. The constraint weight parameters were set in accordance with the operating point number 8 in Table. Analysis of the data in Figure 4 indicates that the network performance varies considerably as the graph instance changes for fixed values of the operating point, the connectivity level, the path length, and the graph size.

19 9 Performance vs. Graph Instance Number of Trials 3 0 [.04,.06) [.06,.08) [.08,.0) [.0,.) [.,.4) [.4,.6) [.6,.8) Convergence Rate Interval Figure 4. Convergence rate vs. graph instance for the GPSP. Simulation studies indicate that the performance of the network depends heavily on the operating point choice, the problem size, the path length, and the connectivity of the graphs. The exhaustive mathematical stability analysis confirmed that the stable equilibrium point set included all solution points and often a significant number of non-solution points. The number of non-solution equilibrium points which are stable depended on values of the operating point, the problem size, the graph connectivity, path length, and the graph instance. Therefore, as a combination of those parameters were changed, the cardinality of the stable equilibrium point set, which necessarily included all solutions, also changed as reflected by the values of the convergence ratio parameter. The convergence ratio showed a good degree of correlation with the convergence rate and helped explain the results obtained through simulation. 3.. Performance vs. Problem Size The performance of the network for graph sizes of up to, and including, 50 vertices was next studied. The previous work indicates the convergence rate is likely to drop significantly as the graph size increases. Thus, the variables (the graph connectivity, the path length and the operating point) are set to maximize the convergence rate so that the network is likely to converge to a solution without requiring unreasonable simulation time. Earlier simulation work shows the convergence rate is the highest for low graph connectivity, the operating point given by number in Table, and low path lengths. The operating

20 0 point used in the testing of 0 through 50 vertex graphs is very close to number in the constraint weight parameter space and defined by number 8 in Table. In order to save computational effort, the tests were run until the network converged to a solution. The simulation was then terminated by recording the number of relaxations attempting to converge to the solution, rather than running the simulation for 00 relaxations. Additionally, not all test cases in the test matrices were implemented since they would, otherwise, require a very large amount of computation. In all the tables that follow, ">00" indicates the network is likely to require more than 00 iterations to locate a stable solution. The symbol " " denotes that a graph with corresponding specifications could not be generated within 00 attempts and thus, no test results are available. The blank boxes represent the cases where no testing was done. The performance analysis of the network configured to solve the 0, 30, 40, and 50-vertex GPSP are presented in Tables 3, 4, 5, and 6, respectively. Simulation data indicates that the network performance degrades significantly as the problem size increases. The network also occasionally manages to locate a solution. Earlier studies on smaller size graphs showed that the size of the stable equilibrium point set increased as the problem size increased, which helped explained the significant drops in the convergence rate value. The network attempted 0 relaxations for the 30-vertex GPSP with a connectivity level of 0.05 and a path length of 7, Table 4, before locating a solution. In other words, it converged to 9 non-solution stable equilibrium points before it was able to converge to a solution. The same network required 73 relaxations for the 40-vertex GPSP with a connectivity of 0.0 and a path length of 5, Table 5, to converge to a solution. The increase in the number of non-solution stable equilibrium points as the problem size grows is most likely to be the reason for the degradation of the network performance.

21 Connectivity Path Length Level > Table 3. Iterations for convergence to a solution for the 0-vertex GPSP. Connectivity Path Length Level > > Table 4. Iterations for convergence to a solution for the 30-vertex GPSP. Connectivity Path Length Level > Table 5. Iterations for convergence to a solution for the 40-vertex GPSP. Connectivity Path Length Level >00 > > Table 6. Iterations for convergence to a solution for the 50-vertex GPSP.

22 4. Conclusions Simulation results show that the network algorithm does not scale well with the increase in the size of the problem. The mathematical stability analysis of equilibrium points in the state space of the problem indicates that the stable point set includes all the solutions as well as some non-solution points. The same analysis also shows that the number of non-solution stable equilibrium points become much larger than the number of solutions in the stable point set as the size of the problem increases. Therefore, the network becomes more likely to converge to non-solution stable equilibrium points as the problem size increases. Even though many non-solution equilibrium points become stable for bounds on constraint weight parameters for solutions to be stable, the size of the stable point set is only a very small fraction of the size of the equilibrium point set. This indicates the established bounds on constraint weight parameters confine the search effort to a very small subspace of the problem state space. The topology of the network and the second-order Liapunov function in its generic form as defined by Hopfield are not adequate to guide the network to a solution during each relaxation. There are a number of areas which might be studied further to improve the existing Hopfield network algorithm. The problem representation and the definition of the problem specific Liapunov (energy) function could be further studied. The idea behind a new and better representation for the problem is to discover a form for the Liapunov function which will result in the two sets of interest being equal, namely the stable point set and the solution set. Results of this work confirmed once more that a quick and average quality solution is the promise of the Hopfield network. Global search algorithms like the simulated annealing and genetic programming are shown to be better for problems where near-optimal solutions are needed []. There are also efforts in the literature in terms of combining the Hopfield network paradigm, which is a local search algorithm, with a genetic programming paradigm, which is a global search algorithm, in order to benefit from desirable features of both paradigms [0]. Acknowledgment We wish to express our appreciation to the anonymous referees of this paper for their valuable feedback.

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